On the Stern sequence and its twisted version

TL;DR

This paper proves conjectures relating the Stern sequence and its twisted version, providing new identities and simplifying proofs for existing results in sequence theory.

Contribution

It offers simple proofs for Bacher's conjectures and identities for variations of the Stern sequence, including Bacher's twisted sequence.

Findings

Proved Bacher's three conjectures about the twisted Stern sequence.

Derived identities similar to Coons' for sequence variations including Bacher's sequence.

Provided simplified proofs for existing identities in sequence theory.

Abstract

In a recent preprint on ArXiv, Bacher introduced a twisted version of the Stern sequence. His paper contains in particular three conjectures relating the generating series for the Stern sequence and for the twisted Stern sequence. Soon afterwards Coons published two papers in {\it Integers}: first he proved these conjectures, second he used his result to obtain a correlation-type identity for the Stern sequence. We recall here a simple result of Reznick and we state a similar result for the twisted Stern sequence. We deduce an easy proof of Coons' identity, and a simple proof of Bacher's conjectures. Furthermore we prove identities similar to Coons' for variations on the Stern sequence that include Bacher's sequence.